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<h2><i><br>

Elements of Finite Geometry</i><br>


<small>by Steven H. Cullinane</small></h2>
<span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: rgb(255, 255, 255) none repeat scroll 0%; font-size: 13px; vertical-align: baseline; color: rgb(51, 51, 51); font-family: 'Helvetica Neue',Arial,sans-serif; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 17px; text-align: left; text-indent: 0px; text-transform: none; widows: 1; word-spacing: 0px; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"></span><span class="note" style="border: 0px none ; margin: 0px; padding: 0px; background: rgb(255, 255, 255) none repeat scroll 0%; font-size: 12px; vertical-align: baseline; color: rgb(102, 102, 102); font-family: 'Helvetica Neue',Arial,sans-serif; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 17px; text-align: left; text-indent: 0px; text-transform: none; widows: 1; word-spacing: 0px; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">A work under construction, last updated Jan. 16, 2016. 
(Downloadable compressed folder last updated Jan. 9, 2016.)<br>
</span>
<ul style="border: 0px none ; margin: 0px; padding: 0px; background: rgb(255, 255, 255) none repeat scroll 0%; font-size: 13px; vertical-align: baseline; list-style-type: disc; list-style-image: none; list-style-position: outside; color: rgb(51, 51, 51); font-family: 'Helvetica Neue',Arial,sans-serif; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: 17px; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 1; word-spacing: 0px; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">
  <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">A
downloadable compressed folder of the source website
finitegeometry.org/sc was last updated on Jan. 9, 2016. Download at
&nbsp;<a href="http://finite-geometry.github.io/galois/finitegeometry-Jan-9-2016.zip">http://finite-geometry.github.io/galois/finitegeometry-Jan-9-2016.zip</a> .</span></li>
  <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">This is a work under construction using WorkFlowy. The Web address to read the preliminary version  is&nbsp;<a href="https://workflowy.com/s/WStyDzkupQ">https://workflowy.com/s/WStyDzkupQ</a> . (Click "Help" for instruction on using folded text.)</span></li>
  <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Introduction</span><br>
    <span class="note" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; color: rgb(102, 102, 102); -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">This
material is intended as a course supplement or independent-study source
for college students (undergraduate and graduate) who have taken
courses in abstract algebra (linear algebra and group theory) and
modern geometry (affine, projective, finite).
It may also interest some professional mathematicians, since the
content has not been published previously except as follows.
A preliminary version was distributed at the 1976 summer AMS-MAA joint
summer meetings in Toronto. Excerpts were published in the
journal&nbsp;Computer Graphics and Art&nbsp;(Vol. 2, No. 1, February
1977, pp. 5-7).
Additional notes were distributed informally, and are available
as&nbsp;Notes on Groups and Geometry, 1978-1986&nbsp;(pdf ). These and
later notes are, at this writing, on the Web at
http://finitegeometry.org/sc/map.html. This book is based on the notes
at that site.</span></li>
  <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Prerequisites</span><br>
    <span class="note" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; color: rgb(102, 102, 102); -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">"There is no royal road to geometry."&nbsp;
-- Saying attributed to Euclid
.<br>
The reader is not expected to fully understand this book unless he or
she has had undergraduate instruction in linear algebra, abstract
algebra, and modern geometry. The following references may, however, be
helpful:
Books:
<span class="contentItalic" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; font-style: italic; line-height: 0pt; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Symmetry</span>, by Hermann Weyl, <span class="contentItalic" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; font-style: italic; line-height: 0pt; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Contemporary Abstract Algebra</span>, by Joseph A. Gallian, and
    <span class="contentItalic" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; font-style: italic; line-height: 0pt; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Geometry and Symmetry</span>, by Paul B. Yale</span></li>
  <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">History</span>
    <ul style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; list-style-type: disc; list-style-image: none; list-style-position: outside; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">
      <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Galois fields</span><br>
        <span class="note" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; color: rgb(102, 102, 102); -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">In 1830, Galois introduced <span class="contentItalic" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; font-style: italic; line-height: 0pt; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">finite</span> (or, now, <span class="contentItalic" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; font-style: italic; line-height: 0pt; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Galois</span>) fields, a new type of number systems that would later prove widely useful. </span></li>
      <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Permutation groups</span><br>
        <span class="note" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; color: rgb(102, 102, 102); -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Galois also introduced permutation <span class="contentItalic" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; font-style: italic; line-height: 0pt; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">groups</span>, a very basic and important concept with which we assume the reader is familiar.</span></li>
      <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Froebel cubes</span><br>
        <span class="note" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; color: rgb(102, 102, 102); -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">
In 1838, educator Friedrich Fr�bel introduced his "third gift," a boxed
set of eight identical cubical &nbsp;unmarked kindergarten blocks.</span></li>
      <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Fano geometries</span><br>
        <span class="note" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; color: rgb(102, 102, 102); -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">In 1892, Gino Fano introduced <span class="contentItalic" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; font-style: italic; line-height: 0pt; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">finite</span>
geometries. These were abstract structures defined axiomatically. They
were unrelated, so far as anyone knew at the time, to any actual <span class="contentItalic" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; font-style: italic; line-height: 0pt; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">physical</span> structures.</span></li>
    </ul>
  </li>
  <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Structures considered</span>
    <ul style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; list-style-type: disc; list-style-image: none; list-style-position: outside; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">
      <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">1. The fourfold square &#8212; A 2x2 array of square unit cells</span></li>
      <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">2. The eightfold cube &#8212; A 2x2x2 array of unit cubes</span></li>
      <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">3. The 16fold square &#8212; A 4x4 array of square unit cells</span></li>
      <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">4. The 24fold rectangle &#8212; A 4x6 array of square unit cells, split into three 4x2 "bricks"</span></li>
      <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">5. The 64fold cube &#8212; A 4x4x4 array of unit cubes</span></li>
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  <li style="border: 0px none ; margin: 4px 0px 4px 20px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;"><span class="name" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 13px; vertical-align: baseline; -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Coordinates</span><br>
    <span class="note" style="border: 0px none ; margin: 0px; padding: 0px; background: transparent none repeat scroll 0%; font-size: 12px; vertical-align: baseline; color: rgb(102, 102, 102); -moz-background-clip: initial; -moz-background-origin: initial; -moz-background-inline-policy: initial;">Coordinates
formed from the two-element Galois field GF(2) may be applied to each
of these in such a way that the resulting labeled structures have large
and interesting symmetry groups. In the case of the 4x6 array, it is
useful to view coordinates in each 4x2 brick not as 3-tuples formed
from GF(2), but rather as elements of the field GF(8).</span></li>
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